A circle is made up of points **equidistant** from the center. But what does “equidistant” mean? Measuring distance implies a value judgment — for example, that moving to the left is just the same as moving to the right, moving forward is just as hard as moving back.

But what if you’re **on a hill**? Then the amount of force to go uphill is different than the amount to go downhill. If you drew a picture of all the points you could reach with a fixed amount of work (**equiforce** or equiwork or equi-effort curve) then it would **look different** — slanted, tilted, bowed — but still be “even” in the same sense that a circle is.

Here’re some **brain-wrinkling pictures of “circles”**, under different L_p metrics:

** ****p** = ⅔

The subadditive “triangle inequality” **A→B→C > A→C **no longer holds when **p<1**.

**p** = 4

p = ½. (Think about a **Poincaré disk** to see how these pointy **astroids** can be “circles”.)** **

** ****p** = 3/2

The moves available to a **knight** ♘ ♞ in **chess** are a circle under **L1 metric** over a **discrete** 2-D space.

### Like this:

Like Loading...

*Related*

## About isomorphismes

Argonaut: someone engaged in a dangerous but potentially rewarding adventure.